A radio transmission tower is 120 feet tall. How long should a guy wire be if it is to be attached 15 feet from the top and is to make an angle of 26 degrees with the ground? Give your answer to the nearest tenth of a foot
Let its length be h
After making a sketch , I can see that
sin 26° = 105/h
h = 105/sin26 = 239.52
or 239.5 ft
Well, it seems like the radio transmission tower wants to take up a new hobby - tightrope walking! Let's help it out.
To find the length of the guy wire, we can use some trigonometry. Specifically, the tangent function (because we have an angle and one side length).
First, we need to find the height of the guy wire. Given that it's attached 15 feet from the top of the tower, the height of the guy wire is the tower's height minus 15 feet. So, the height is 120 - 15 = 105 feet.
Now, let's calculate the length of the guy wire using the tangent function:
tangent(26 degrees) = opposite / adjacent
The opposite side is the height of the guy wire (105 feet), and the adjacent side is what we're looking for, the length of the guy wire.
Therefore, the equation is:
tan(26 degrees) = 105 / length
To solve for the length, we rearrange the equation:
length = 105 / tan(26 degrees)
Using a calculator, we find that the length of the guy wire is approximately 211.9 feet.
So, the guy wire should be about 211.9 feet long. Keep in mind that this answer is rounded to the nearest tenth of a foot, making it 211.9 feet.
To find the length of the guy wire, we can use the trigonometric function tangent.
First, we need to determine the opposite side length, which is the height difference between the tower and the attachment point of the guy wire:
Opposite side = tower height - attachment point height
Opposite side = 120 ft - 15 ft
Opposite side = 105 ft
Next, we can use the tangent function to find the length of the guy wire:
Tangent(angle) = opposite / adjacent
In this case, the angle is 26 degrees, the opposite side is 105 ft, and the adjacent side is the length of the guy wire.
Tangent(26 degrees) = 105 ft / adjacent
To find the length of the adjacent side (guy wire), we need to rearrange the equation:
adjacent = opposite / tangent(angle)
adjacent = 105 ft / tangent(26 degrees)
Using a scientific calculator or trigonometric table, we can find that the tangent of 26 degrees is approximately 0.4877.
adjacent = 105 ft / 0.4877
adjacent ≈ 215.1 ft
Therefore, the length of the guy wire should be approximately 215.1 feet (rounded to the nearest tenth of a foot).
To calculate the length of the guy wire, we can use trigonometry. Specifically, we'll use the sine function (sin) because we know the angle and the length from the top of the tower to where the guy wire is attached.
Let's break down the problem step by step:
1. Draw a diagram: Sketch a right-angled triangle with the radio transmission tower as one side, the guy wire as the hypotenuse, and the vertical line connecting the top of the tower to where the guy wire is attached.
|
|
T |--------------
| |
| |
| | G
| |
| |
| |
| |
/|\
/ | \
/ | \
/ | \
/ | \
/ | \
T = height of the tower
G = length from the top of the tower to the guy wire attachment point
2. Identify the given values:
- Height of the tower (T) = 120 feet
- Distance from the top (G) = 15 feet
- Angle with the ground = 26 degrees
3. Determine the missing side using trigonometry:
We are trying to find the length of the guy wire (H).
Since we have the opposite side (T) and the adjacent side (G), we can use the formula:
sin(angle) = opposite/hypotenuse
sin(26 degrees) = T/H
Rearrange the formula to isolate H:
H = T / sin(26 degrees)
4. Calculate the length of the guy wire:
H = 120 feet / sin(26 degrees)
Using a scientific calculator, we can find the sine of the angle, which is approximately 0.4384.
H = 120 feet / 0.4384
H ≈ 273.58 feet
Therefore, the length of the guy wire should be approximately 273.6 feet when rounded to the nearest tenth of a foot.